Darboux-Backlund Transformation Research Articles

Darboux–Bäcklund transformation and explicit solutions to a hybrid lattice of the relativistic Toda lattice and the modified Toda lattice

Abstract The relativistic Toda lattice and the modified Toda lattice are two important discrete integrable equations. In this paper, we investigate a hybrid lattice equation of the two lattice equations. Darboux–Backlund transformation and explicit solutions to the hybrid lattice equation are constructed.

Dispersion and nonlinearity managed multisoliton propagation in an erbium doped inhomogeneous fiber with gain/loss

We propose a system of coupled generalized nonlinear Schrodinger and Maxwell-Bloch equations with variable coefficients which describes propagation of an optical pulse in an inhomogeneous fiber doped with two level resonant atoms. The Lax pair is presented and multi-soliton solutions are constructed through a Darboux-Backlund transformation. We explain soliton controls for various cases like distributed amplification, pulse compression etc. with varying group velocity dispersion, nonlinearity and gain/loss parameters.

The Darboux–Bäcklund transformation for the static 2-dimensional continuum Heisenberg chain

We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix depending on the background solution. In order to obtain the transformation we use a new, more general, spectral problem.

Geometry of Submanifolds Derived from Spin-Valued Spectral Problems

We present recent results motivated by Sym's theory of soliton surfaces. Quite general assumptions about the structure of the spectral problem can lead to some specific classes of surfaces. In some cases (including pseudospherical surfaces), this approach is coordinate-independent, which seems a surprising novelty. The Darboux–Backlund transformation is formulated in terms of Clifford numbers, which greatly simplifies constructing explicit solutions. Cumbersome computations in matrix representations are replaced with rotations represented by elements of an appropriate Spin group. Finally, the spectral problem and the spectral parameter are derived purely geometrically in the case of isometric immersions of constant-curvature spaces in spheres and Euclidean spaces.

Solitons ofq-deformed quantum lattices and the quantum soliton

We use the classical N-soliton solution of a q-deformed lattice, the Maxwell-Bloch (MB) lattice, which we reported recently (Rybin A V, Varzugin G G, Timonen J and Bullough R K Year 2001 J. Phys. A: Math. Gen. 34 157) in order, ultimately, to fully comprehend the `quantum soliton'. This object may be the source of a new information technology (Abram I 1999 Quantum solitons Phys. World 21-4). We suggested in Rybin et al 2001 that a natural quantum mechanical matrix element of the q-deformed quantum MB lattice becomes in a suitable limit the classical 1-soliton solution of the classical q-deformed MB lattice explicitly derived by a variant of the Darboux-Backlund method. The classical q-deformed MB lattice was introduced in Bogoliubov N M, Rybin A V, Bullough R K and Timonen J 1995 Phys. Rev. A 52 1487. In this short paper the quantum inverse method is viewed as a Darboux-Backlund transformation at quantum level, two q-deformed quantum lattices are introduced and solved, and relevant matrix elements are formally derived. Further investigation of the classical limits of these matrix elements must however be deferred until future work.

On Darboux-Bäcklund Transformation and Positon Type Solution of Coupled Nonlinear Optical Waves

A Darboux-Backlund transformation is used to obtain a positon type solution of the nonlinear equations describing the propagation of coupled nonlinear optical pulses.This form of the positon solution is then compared with that obtained by the special limiting procedure applied to a two-soliton solution. It is observed that though the algebraic form of the two solutions is different yet both of these have singularities and the position of the singularities remains on the similar curve in the (x, t) plane. We also depict the form of these solutions graphically. Finally, it may be added that the method of Darboux-Backlund transformation is convenient for generating more than one-positon solution. PACS numbers: 02.30.+g, 52.35.Sb, 52.35.Mw

Complete integrability ofN-coupled higher-order nonlinear Schrödinger equations in nonlinear optics

The coupled higher-order nonlinear Schrodinger equations are generalized to an N-coupled system which governs the propagation of N fields in a fibre with all the higher-order effects like third-order dispersion, Kerr dispersion and stimulated Raman scattering. The explicit Lax pair is constructed and the soliton solutions are obtained using the Darboux-Backlund transformation. Finally, the Hirota bilinear form for the N-coupled system equations is also presented.

A note on the gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies

The gauge equivalence between the Manin-Radul and Laberge-Mathieu super KdV hierarchies is revisited. Apart from the Inami-Kanno transformation, we show that there is another gauge transformation which also possess the canonical property. We explore the relationship of these two gauge transformations from the Kupershmidt-Wilson theorem viewpoint and, as a by-product, obtain the Darboux-Backlund transformation for the Manin-Radul super KdV hierarchy. The geometrical intepretation of these transformations is also briefly discussed.

Optical Soliton Propagation in a Coupled System of the Nonlinear Schrödinger Equation and the Maxwell-Bloch Equations

Optical soliton propagation in fibres with resonant impurities and erbium-doped nonlinear systems are considered. The fibre system is governed by the coupled system of the nonlinear Schrodinger equation and the Maxwell-Bloch equations. The system equations using Painleve analysis are found to be completely integrable, but only when the parameters involving the group velocity dispersion, Kerr nonlinearity and the field interaction with the atoms, satisfy a particular condition. With a suitable approximation, for the first time Painleve analysis of the system equation is reported in this paper. The Lax pair of the system equation is explicitly constructed. The single soliton solution is also explicitly derived using the Darboux-Backlund transformation method.

Unified approach to the conservation laws and soliton solution for sine-gordon system

It is possible to generate an infinite number of conserved quantities and the most general soliton solution in an arbitrary background with the help of the Darboux-Backlund transformation and an expansion of the Lax eigenfunction in the eigenvalue parameter. Use is not made of the Riccati form of the Lax equation which is used in the usual derivation of the conserved quantities. It is shown that for zero seed solution one retrieves the usual one-soliton solution.

INVERSE SCATTERING TRANSFORM AND REGULAR RIEMANN-HILBERT PROBLEM

It is shown that the inverse scattering transform method to solve the Lax pair of given nonlinear evolution equation can be reduced to a kind of Riemann-Hilbert (R-H) problem of meromorphic functions with respect to the complex spectral parameter. The R-H problem is generally regular no matter whether the solitons are involved in it. The linear singular integral equation connected with the R-H problem has been derived, which is essencially equivalent to the Gel'fand-Levitan-Marchenko equation. Furthermore, the regular R-H problem satisfied by the Darboux-B?cklund transformation from a fundamental solution set of the eigenvalue equation of Lax pair to a new set has been given as well. The R-H problem reduced from the inverse scattering transform is in fact a special case of that satisfied by the Darboux-Backlund transformation.